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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Wallman-type compactifications on 0-dimensional spaces


Author: Li Pi Su
Journal: Proc. Amer. Math. Soc. 43 (1974), 455-460
MSC: Primary 54D35
DOI: https://doi.org/10.1090/S0002-9939-1974-0339079-9
MathSciNet review: 0339079
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Abstract: Let $ E$ be Hausdorff 0-dimensional, $ \mathcal{D}$ the discrete space $ \{0, 1\}$, and $ \mathcal{N}$ the discrete space of all nonnegative integers. Every $ E$-completely regular space $ X$ has a clopen normal base $ \mathcal{F}$ with $ X\backslash F \in \mathcal{F}$ for each $ F \in \mathcal{F}$. The Wallman compactification $ \omega (\mathcal{F})$ is $ \mathcal{D}$-compact. Moreover, if an $ E$-completely regular space $ X$ has a countably productive clopen normal base $ \mathcal{F}$ with $ X\backslash F \in \mathcal{F}$ for each $ F \in \mathcal{F}$, then the Wallman space $ \eta (\mathcal{F})$ is $ \mathcal{N}$-compact. Hence, if $ X$ has such an $ \mathcal{F}$, and is an $ \mathcal{F}$-realcompact space, then $ X$ is $ \mathcal{N}$-compact.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0339079-9
Keywords: Wallman spaces, Wallman compactifications, $ E$-completely regular, $ \mathcal{D}$-compact, $ \mathcal{N}$-compact, 0-dimension, complemental base
Article copyright: © Copyright 1974 American Mathematical Society

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